Hot-Electron Effect in Superconductors and Its ... Effect in Superconductors and Its Applications for ... tions describing the hot-electron effect in superconductors ... 100-fs optical

HOT-ELECTRON EFFECT IN SUPERCONDUCTORS AND ITS APPLICATIONS FOR RADIATION SENSORS

134 LLE Review, Volume 87

IntroductionThe term “hot electrons” was originally introduced to describenonequilibrium electrons (or holes) in semiconductors (for areview see, e.g., Ref. 1). The term encompasses electrondistributions that could be formally described by the Fermifunction but with an effective elevated temperature. The con-cept is very fruitful for semiconductors, where the mobility ofcarriers can be shown to depend on their effective temperature.In metals, however, electrons do not exhibit any pronouncedvariation of the mobility with their energy. As a result, heatingof electrons in a metal does not affect the resistance,2 unless thechange in the effective temperature is comparable with theFermi temperature.

Schklovski3 was the first to discuss the idea of combiningthe steady-state electron heating with the strong dependence ofthe resistance on the effective electron temperature in a metalfilm undergoing the superconducting transition. In the steady-state regime, however, electron heating is always masked bythe conventional bolometric effect; therefore, experimentalresults on the heating of electrons by the dc current were notvery convincing. The regime of dynamic electron heating byexternal radiation was studied in a series of experimental andtheoretical works.4–6 It was immediately realized that the veryshort relaxation time of electron excitations would make itfeasible to design extremely fast radiation sensors with asensitivity much better than that of conventional bolometers.

During the last decade, a new generation of hot-electronsuperconducting sensors has been developed. These includesubmillimeter and THz mixers, direct detectors, and photoncounters for the broad spectral range from microwaves tooptical radiation and x rays. Activity in the field of hot-electronsuperconducting sensors is growing rapidly. These sensorshave already demonstrated performance that makes them de-vices-of-choice for many far-infrared (THz), infrared, andoptical wavelength applications, such as plasma diagnostics,laser studies, ground-based and airborne heterodyne astronomy,and single-photon-detection and quantum communications.Parallel development of compact cryocoolers and THz radia-

Hot-Electron Effect in Superconductors and Its Applicationsfor Radiation Sensors

tion sources opens hot-electron sensors for satellite astronomyand communication applications. This article reviews thephysical background of the hot-electron phenomenon in super-conducting films and discusses various technical realizationsof hot-electron radiation sensors.

Physics of Hot ElectronsThermal dynamics in a superconducting film on a dielectric

substrate can be thought of in terms of four co-existing sub-systems: Cooper pairs, quasiparticles (electrons from brokenCooper pairs), phonons in the film, and phonons in the sub-strate. Thermal equilibrium exists when all of these can bedescribed by equilibrium distribution functions with the sametemperature. If any distribution does not satisfy these condi-tions, the situation is considered nonequilibrium. Generaltreatment of a nonequilibrium state requires solution of theintegral kinetic equations for space- and time-dependent distri-bution functions. To avoid the above complexity, varioussimplifying assumptions are used to reduce the general prob-lem to analytically solvable rate equations.

1. Hot-Electron Cooling and DiffusionThe hot-electron model is most relevant for nonequilibrium

superconductors maintained at temperature T near the super-conducting transition Tc, where quasiparticles and phononscan be described by thermal, normal-state distribution func-tions, each with its own effective temperature. The electronand phonon effective temperatures (Te and Tp) are assumed tobe established instantly and uniformly throughout the wholespecimen. This assumption implies that a rapid thermalizationmechanism exists inside each subsystem.

The main steps of the hot-electron phenomenon that lead tothe global equilibrium are depicted in Fig. 87.27. Introducingcharacteristic times of the energy exchange between sub-systems reduces the problem of the global equilibrium recov-ery to a pair of coupled heat-balance equations for Te and Tp.The intrinsic thermalization time τT should be short comparedto energy exchange times. This two-temperature (2-T) ap-proach was used for the first time by Kaganov et al.2 to describe


LLE Review, Volume 87 135

steady-state electron heating in metals. Below Tc, the electronspecific heat exhibits an exponential temperature dependencethat makes equations nonlinear for even small deviations fromequilibrium. The description can, however, be simplified in thevicinity of Tc. At this temperature the superconducting energygap is strongly suppressed, concentration of Cooper pairs isvery small, and unpaired electrons exhibit no significant super-conducting peculiarities: they are regarded as normal electronshaving the ordinary Fermi distribution function. In the normalstate, the specific heat of electrons has a much weaker tempera-ture dependence, which can be neglected for small deviationsof Te from the equilibrium. With these assumptions, the equa-tions describing the hot-electron effect in superconductorsbecome linear and can be written as

d T

dt

T T

CW te e p

e= −

−+ ( )

τ ep

1, (1a)

d T

dt

C

C

T T T Tp e

p

e p p=−

−−

τ τep es

0 , (1b)

where W(t) represents the external perturbation (i.e., the powerper unit volume absorbed by the electron subsystem); τep andτes are the electron energy relaxation time via electron–phononinteraction and the time of phonon escape into the substrate; Ceand Cp are the electron and phonon specific heats, respectively;

Figure 87.27Thermalization scheme showing various channels of the energy transfer in ahot-electron device that relaxes toward global equilibrium.

and T0 is the ambient (substrate) temperature. To derive the2-T equations we used the condition of the energy-flow bal-ance in equilibrium τ τpe ep= ( )C Cp e , where τpe is thephonon–electron energy relaxation time.

The first implementation of the electron-heating model tosuperconductors was made by Shklovski,3 who used a moregeneral, nonlinear form of the heat-balance equations to de-scribe hysteresis of the critical current in a thin lead film. Ananalytical solution of Eq. (1) was first obtained for sinusoidalperturbations by Perrin and Vanneste4 and for an optical pulseexcitation by Semenov et al.5 In the latter case, thermalizationof electrons was interpreted as an increase of Te. The increasewas assumed to occur during a time interval that depended onboth the duration of the optical pulse and the intrinsic thermal-ization time τT. The model was used to describe the responseof superconducting NbN and YBa2Cu3O7−δ (YBCO) films inthe resistive state to near-infrared and visible radiation.5,7

Figures 87.28 and 87.29 show a good agreement betweenexperimental signals and the theoretical simulation.

Frequency (MHz)

Nor

mal

ized

mix

er s

igna

l (dB

)

Z2485

10–1 100 101 102 103 104 105–50

–40

–30

–20

–10

0

10

Figure 87.28Response of a YBCO hot-electron photodetector (HEP) to optical radiation(dots) versus modulation frequency (Ref. 7). The solid line was calculatedusing Eqs. (1). The discrepancy at low frequencies is due to phonon diffusionin the substrate that was not accounted for in the model. The dashed linerepresents the thermal model.

Figure 87.30 presents the detailed thermalization diagramsfor both YBCO [Fig. 87.30(a)] and NbN [Fig. 87.30(b)] thinfilms exited by 100-fs-wide optical pulses. The diagramsdepict the process in the same manner as Fig. 87.27 but nowinclude the actual values of the characteristic time constantsfor both materials. The values were obtained from the 2-Tmodel via the fit of Eqs. (1) to the experimental photoresponse

Substrate Tb

ElectronsCe, Te

teptpe

PhononsCp, Tp

tes

tee

e

w

tpe >> tep

tpe tes

ee

Z2429

pp

pp



Figure 87.29Response of a YBCO HEP to a femtosecond infrared pulse: experimental data(solid line) (Ref. 5) and simulations (dashed line) based on the 2-T model.

data. The measurements were performed using the electro-optic sampling system, which allowed obtaining the intrinsic,time-resolved dynamics of the electron thermalization processin 3.5-nm-thick NbN8 and 100-nm-thick YBCO films.9 Wenote that, in general, the dynamic of the YBCO thermalizationis roughly one order of magnitude faster than that of NbN. Inboth cases, the energy flow from electrons to phonons domi-nates the energy backflow due to reabsorption of nonequilibriumphonons by electrons; however, while the energy backflow inYBCO can be neglected because of the very large ratioC Cp e = 38 , in NbN it constitutes a non-negligible 15%C Cp e =( )6 5. of direct electron–phonon energy relaxation.

Consequently, in YBCO film excited on the femtosecond timescale, the nonthermal (hot-electron) and thermal, bolometric(phonon) processes are practically decoupled, with the formertotally dominating the early stages of electron relaxation. Onthe other hand, the response of NbN devices is determined bythe “average” electron cooling time τe, which is given byτ τep es+ +( )1 C Ce p

4,5 and corresponds to the time that elapsesfrom the peak response until the magnitude of the responsedeclines to 1/e of the maximum value. If the external perturba-tion is substantially longer than τpe (that is, >100 ps for YBCOfilms), the YBCO response is dominated by the bolometricprocess, as was shown by the bulk of the early photoresponsemeasurements.10 The very large difference in the τes values forYBCO and NbN is mainly due to the drastic difference inthickness of the tested films. Additionally, ultrathin NbN filmsare a better acoustical match to the substrate. This significantlyreduces τes.

Electron heating in the limiting case of a very short phononescape time, τes << τep, τpe, was first studied by Gershenzonet al.6 for Nb films. Although for this material11 C Cp e ≈ 0 25.and, consequently, τep > τpe, the effective escape of phonons to

Figure 87.30Hot-electron relaxation diagrams and characteristic times for (a) thin-filmYBCO (Ref. 9) and (b) ultrathin NbN film (Ref. 8).

the substrate prevents energy backflow to electrons. As aresult, τep alone controls the response of ultrathin (<10-nm)Nb films. Typical electron relaxation time in Nb is ≈1 ns at4.2 K, which is over an order of magnitude larger than in NbN.

The 2-T model represented by Eqs. (1) is essentially thesmall-signal model. Deviations of the effective temperaturesfrom equilibrium due to both the joule power dissipated by thebias current and absorbed radiation power are assumed smallcompared to their equilibrium values. The theory of operationof a hot-electron photodetector (HEP) was developed on thebasis of this model by Gershenzon et al.,12 and a novel hot-electron mixer (HEM) was proposed.12,13

1 ns

Time

Res

pons

e

Z2486a Z2557

ElectronsCe, Te

PhononsCp, Tp

Substrate Tb

tep ≈ 10 ps

tes = 38 ps

e

tpe = 6.5 tep

p

pp

p

ee

100-fs optical pulse

tT = 6.5 ps

tpe = 1.7 tes

ElectronsCe, Te

PhononsCp, Tp

Substrate Tb

tep = 1.1 ps

tes = 3.5 ns

e

tpe = 38 tep = 42 ps

tes >> tpe

p

pp

p

ee

100-fs optical pulse

tT = 0.56 ps

(a)

(b)



The 2-T approach neglects, however, diffusion of electronsand assumes that the effective temperatures remain uniformwithin the whole device. A different approach was proposed byProber,14 who considered diffusion of hot electrons out of theactive area, rather than the energy transfer to phonons, as themain mechanism of the electron cooling. If the device lengthL is short compared to the thermal diffusion length Lth =(Dτe)

1/2, where τe is the electron cooling time and D is theelectron diffusivity, relaxation of Te is controlled by the elec-tron out-diffusion time τ πd L D= ( )2 2 . In the limiting caseL << Lth, Te remains almost uniform through the device length.The device can then be described by Eq. (1a), in which τep andTp should be substituted for τd and T0, respectively. For longerdevices, both the actual distribution of Te along the devicelength and the phonon contribution to the electron relaxationshould be taken into account.

2. Large-Signal ModelsThe common disadvantage of the small-signal model de-

scribed above is that the optimal values of the bias current (for

Figure 87.31(a) Current–voltage characteristics fordifferent LO power values. (b) Conver-sion gain curves for a NbN HEM com-pared with results of the uniform model(solid lines) (Ref. 15).

HEP’s) and power of the local oscillator (for the HEM theory)are not derived in the framework of the model, but rather takenfrom the experiment or independently estimated. To includethe bias current and the local oscillator (LO) power in a con-sistent manner, one should specify the structure of the resistivestate and account for the dependence of the electron-coolingrate on the deviation from the equilibrium. For large deviationsfrom equilibrium, heat-balance equations become nonlinear.

The large-signal mixer theory was developed by Nebosiset al.15 for the uniform resistive state (which is, of course, avery crude approximation). The authors assumed a finite valueof τes and introduced the superconducting critical current.Reasonable quantitative agreement (see Fig. 87.31) was foundbetween the experimental data for NbN mixers and the theo-retical results. Karasik et al.16 implemented a similar approachfor modeling a bolometric mixer fabricated from a high-temperature superconducting material. Floet et al.17 consid-ered the nonuniform resistive state of a hot-electron bolometerin the small-signal regime for τes = 0, while Merkel et al.18

developed the large-signal nonlinear model for a finite, non-zero value of τes. Both models described the resistive state ofthe mixer at optimal operation conditions in terms of a normalhot spot, maintained by self-heating. The hot spot occupiesonly a portion of the device length, thus assuring a mixerresistance between zero and the normal-state value. In thisapproach, the LO power is assumed to be uniformly absorbedin the mixer, whereas the joule power dissipation due to thebias current appears in the hot-spot region only. Since thediffusion of electrons is introduced in the basic equations, thismodel naturally covers all intermediate cases between theextreme diffusion cooling (L << Lth) and phonon-cooling(L >> Lth) regimes. Neglecting phonons (τes = 0) and simulta-

Current (mA)

Con

vers

ion

gain

(dB

)

20 30 40 50 60

–30

10

(b)

–34

–32

PLO = 130 mW

Z2487

Voltage (mV)

Cur

rent

(mA

)

2 4

20

40

60

124 mW130 mW135 mW143 mW

0

(a)

0

–28

–26

–24

LO power (mW)

Con

vers

ion

gain

(dB

)

120 140 160 180

–30

–32

I = 38 mA

–28

–26

6



neously assuming τT = 0, one can reduce the problem to thefollowing system of equations17 for Te:

− + −( ) = + ( )

− + −( ) = ( )

Kd T

dx

CT T j P

Kd T

dx

CT T P

e e

ee n

e e

ee

2

2 02

2

2 0

τρ

τ

RF

RF

inside hot spot

outside hot spot

,

,

(2)

where K is the thermal conductivity, j is the bias currentdensity, ρn is the resistivity of the mixer in the normal state,and PRF is the LO power absorbed per unit volume. Thisdescription allows for an analytical solution, which returns thebias current as a function of the hot-spot length, and, thus, avoltage drop across the device. Results of simulations17 arein good agreement with the experimental current–voltage(I–V) characteristics, especially for large PRF values, whichdrive the mixer almost into the normal state. Surprisingly,results based on not only Eq. (2), but even on the more-accuratenumerical model18 shown in Fig. 87.32, do not differ muchfrom simulations based on the uniform 2-T model (Fig. 87.31).With the appropriate set of fitting parameters, both approachesdescribe fairly well the I–V characteristics of the HEM andpredict reasonable values of the conversion efficiency andnoise temperature.

A nonthermal regime of the diffusion-cooled HEM wasdescribed by Semenov and Gol’tsman.19 The authors consid-ered a short device made from a clean material, in which τTis larger than τd. The device operated in the nonthermal regime

Figure 87.32(a) Current–voltage characteristicsand (b) conversion gain of a NbNHEM simulated in the framework ofthe hot-spot model (Ref. 18). Ex-perimental characteristics are shownfor comparison.

and had the advantage of a short response time (or, equiva-lently, a large bandwidth) in the heterodyne mode. On the otherhand, incomplete thermalization hampered the responsivityand increased the relative contribution of the Johnson noise tothe total electric noise of the device. Compared to HEM’s oper-ated in the thermal regime, the nonthermal mixer requiredmore power from LO. At low temperatures, however, thenonthermal regime of operation provided almost quantum-limited sensitivity.

The electric noise of a hot-electron sensor is comprised ofthe same components as the noise of any conventional bolom-eter: shot noise, Johnson noise, thermal noise, and flickernoise. To our knowledge, there is no consistent theory forflicker noise, so its contribution may be determined onlyexperimentally. Unless the sensitivity of the bolometer reachesthe quantum limit, the noise due to fluctuations in the back-ground radiation can be neglected. The typical length of hot-electron devices studied so far was much larger than thediffusion length associated with the electron–electron scatter-ing. In this limiting case, the superposition of Johnson noiseand shot noise reduces to the Nyquist form, i.e., the spectraldensity of the voltage noise is SV = 4 kBT R, where R is theresistance of the device. This noise has a “white” spectrum upto very high frequencies. The corresponding contribution tothe system-noise temperature in the heterodyne regime in-creases rapidly when the conversion efficiency rolls off atintermediate frequencies (IF’s) larger than 1/τe.

Thermal noise contributes to the total spectral density theamount 4 2 2 2

k T I R T CB e e eτ ∂ ∂( ) ( )v , where I is the biascurrent and v is the volume of the sensor. Since the conversionefficiency is proportional to I P R T Ce e e

− ∂ ∂( ) ( )2 2 2 2τ RF v and

Bias voltage (mV)

Con

vers

ion

gain

(dB

)

Z2488

2 3 4 5

–25

–20

–15

–10

–5

0

200 nWf = 1 THzV = 0.8 mVR = 26 ΩLO = 150 nWS = 11 Ω/nWQ = 39 Ω/nW

Bias voltage (mV)

Bia

s cu

rren

t (m

A)

0.2 0.4 0.6 0.8 1.0

20

30

40

50

60

0.0

9.50 K

Theoretical9.33 K

9.66 K

10.02 K

0

(a) (b)

10

Measured

100 nW

50 nW

1

PLO = 20 nW



has the same roll-off frequency, the noise temperature ofthe mixer due to thermal fluctuations is given byT T C PN e e e≈ ( )2 v α τ RF , where α is the optical coupling effi-ciency. The contribution to the noise temperature due tothermal fluctuations does not depend on the intermediatefrequency; neither does the corresponding noise-equivalentpower (NEP) in the direct-detection mode,

NEP ≈ ( )( )T k Ce B e eα τv 1 2.

On the contrary, the contribution due to the Nyquist termincreases rapidly at IF’s larger than 1/τe and usually limits theIF noise bandwidth of the mixer.

Though the above simple treatment of the bolometer noiseexplains the main features, it does not provide an appropriatetool for computations. To obtain exact results, one should takeinto account the positive feedback via the load resistor and self-heating by the bias current. The former enhances the systemoutput noise because the bolometer rectifies part of its ownnoise voltage drop across intrinsic resistance. The latter effecttypically increases the IF bandwidth in the heterodyne regimeand decreases the response time in the direct-detection mode.It is of little practical use, however, because operation in thevicinity of the thermal roll-off requires very precise stabiliza-tion of the ambient temperature. For a HEM with dc resistanceR at the operation point and connected to the IF load with theimpedance RL, the dependence of the conversion efficiencyη(ω) and single-sideband noise temperature TSSB(ω) on the IFwas derived in the framework of the uniform model15

η ω α

ξ ϕ

( ) =+( ) −

++

+∞

∞

22 2

2

22I

R

R R

C P

CR R

R R

L

L L

L

RF , (3)

TT R I

C P

T

C V Pe e e

eSSB

RF RFω

αξ ϕ τ

α( ) = +( ) +∞2 22

22 2

2, (4)

where

C IR T

C Vee

e= ∂ ∂2τ ,

ξ ωω τ τ τ τ τ τ

ω τ( ) =

+ + −( )+ ( )

1

1

21 3 2 3 1 2

32 ,

ϕ ωω τ τ τ ω τ τ τ

ω τ( ) =

+ −( ) +

+ ( )1 2 3

31 2 3

32

1,

ττ τ τ1 2

1 1

2

1 1,

− = + +

ep ep es

C

Ce

p

× ±

+ +

1

4

1 12

τ τ

τ τ τ

ep es

ep ep es

C

Ce

p

,

and

ττ τ

τ3 = ep es

e.

In the above equations, R∞ is the impedance of the bolom-eter at very high IF, and ∂R/∂Te is the slope of the superconduct-ing transition at the operation point on the scale of the electrontemperature. The slope of the transition cannot be derived fromfirst principles in the framework of the uniform model. Itstemperature dependence should be calculated in a phenomeno-logical manner (see, e.g., Ref. 15), or the value at the specificoperation regime should be concluded from the experiment.Ekström et al.20 showed that the magnitude of the parameter Cin Eqs. (3) and (4) can be determined from the experimental dcI–V characteristic as

C

dVdI

R

dVdR

R=

−

+, (5)

where dV/dI is the differential resistance of the HEM at theoperation point. The advantage of the hot-spot model18 is thatit allows for numerical computation of the superconducting



transition slope for arbitrary values of the LO power, biascurrent, and ambient temperature.

3. Cooper-Pair, Kinetic–Inductive PhotoresponseAlthough the response of a superconductor that is kept well

below Tc to external radiation cannot be adequately treated inthe framework of the hot-electron approximation, we decidedto include superconducting detectors operating at T << Tc inour review. Rothwarf and Taylor21 were the first to success-fully develop the phenomenological description for non-equilibrium Cooper-pair recombination and breaking pro-cesses (so-called the RT model). At low temperatures, whenenergies of nonequilibrium quasiparticles after thermalizationare spread over a narrow interval above the superconductingenergy gap 2∆, the appropriate parameters to characterize thisnonequilibrium state are the number ∆nq of excess quasipar-ticles and the number ∆np of excess, so-called, 2∆ phonons.The 2∆ phonons are emitted in the Cooper-pair recombinationprocess and, since they have the energy of at least 2∆, they areresponsible for secondary breaking of Cooper pairs. For smallperturbations, concentrations of ∆nq and ∆np are given by thefollowing linearized RT rate equations:

d

dtn

n nq

q

R

p

B∆

∆ ∆= − +

τ τ2

, (6a)

d

dtn

n n np

p

B

p q

R∆

∆ ∆ ∆= − − +

τ τ τes

2, (6b)

where τR and τB are the quasiparticle recombination time andthe time of breaking Cooper pairs by 2∆ phonons, respectively.We note that Eqs. (6) are mathematically analogous to the 2-Tmodel [Eqs. (1)]. Like the 2-T model, the RT approach assumesthat there is a quick, intrinsic thermalization mechanism insideboth the quasiparticle and phonon subsystems.

When photons with energy typically much larger than2∆ are absorbed by a superconducting film maintained atT << Tc, they produce a time-dependent population ∆nq(t)of nonequilibrium quasiparticles, leading to a temporarydecrease in the superconducting fraction of electrons,f n n nqsc = −( )0 0 , where n n n tq q q= ( ) + ( )0 ∆ is the instant

concentration of the quasiparticles, nq(0) is their equilibriumconcentration, and n0 is the total concentration of electrons.Because the pairs are characterized by non-zero inertia, thisprocess can be modeled as time-varying kinetic inductance:22,23

L tL

fkinkin

sc( ) = ( )0

, (7)

where L dLkin 0 02( ) = ( )µ λ is the equilibrium value per unit

area of the film, λL is the magnetic penetration depth, and d isthe film thickness. The change in time of Lkin in a current-biased superconducting film leads to a measurable voltagesignal across the film edges.

For the limiting case of very fast thermalization, i.e., whenτT is small compared to both τR and τB, the kinetic-inductiveresponse was described by Semenov et al.24 as the product ofthe analytical solution of Eqs. (6) and a fitting factor exponen-tially growing in time. The latter parameter corresponded tothe multiplication cascade of quasiparticles during thermaliza-tion. The kinetic-inductive model describes well the experi-mental results obtained with pulsed and modulated cwexcitations, for both the low-temperature superconductor (LTS)films (Fig. 87.33 and Ref. 24) and the high-temperature super-conductor (HTS) films (Fig. 87.34 and Refs. 9 and 25).

4. Single-Photon-Detection MechanismsSo far this discussion has been limited to integrating detec-

tors in which the energy of a large number of absorbed photonsis distributed among an even larger number of elementarythermal excitations in the detector. That is, individual photonscannot be distinguished, and only the average radiation powerabsorbed by the detector is measured. In the particular case ofa thermal detector, e.g., a bolometer or a hot-electron detectornear Tc, this average absorbed radiation power corresponds toenhanced effective temperatures of phonons and electrons,respectively. In a quantum (photon) detector, a single photoncreates excitations that are collected and counted before theyrelax and before another photon is absorbed. Thus, the detectorregisters each absorbed photon, while the number of collectedexcitations measures the energy of absorbed photons.

The hot-electron quantum detector was first proposed byKadin and Johnson.26 In this model, a photon absorbed some-where in the film initiates a growing hot spot. The resistanceinside the hot spot is larger than in the surrounding area. Evenif the size of the hot spot is much smaller than the size of thefilm, the voltage drop across the current-biased film “feels” thepresence of the hot spot. The disadvantage of this approachfor practical devices stems from the fact that the film isoperated near its Tc and can withstand only a very small currentdensity without being driven into the normal state. Since the



Figure 87.33(a) Conversion gain and (b) signal response of a NbN HEP to pulsed andmodulated cw optical radiation in comparison with model simulations basedon Eqs. (6) (Ref. 24).

Figure 87.34Experimental response (dots) of a YBCO HEP to 100-fs-wide optical pulses(Refs. 9 and 25). Simulated transients were obtained (a) with the uniform hot-electron model [Eqs. (1)] for the operation in the resistive state and (b) withthe RT model [Eqs. (6)–solid line] and the 2-T model [Eqs. (1)–dashed line],for operation at low temperatures in the superconducting state. Inset in (a)shows a bolometric response.

detector response is proportional to the bias current, the smalloperating current requires a complicated, SQUID-based read-out scheme.27

Semenov et al.28 proposed a different quantum detectionregime in a superconducting stripe that is operated well belowTc and carries a bias current only slightly smaller than thecritical value at the operating temperature. Generation of a hotspot at the position where the photon has been absorbed createsa local region with suppressed superconductivity (normalregion). The supercurrent is forced to flow around the normal

(resistive) spot, through those parts of the film that remainsuperconducting. If the diameter of the resistive spot is suchthat the current density in the superconducting portion of thefilm reaches the critical value, a resistive barrier is formedacross the entire width of the stripe, giving rise to a voltagepulse with the magnitude proportional to the bias current.

The physical difference of the quantum detection proposedin Ref. 28, as compared to Ref. 26, is that the resistive state and,thus, the response appear to be caused by the collaborativeeffect of the bias current and the hot-spot formation. In the hot

Z2494

Time (ns)

Con

vers

ion

gain

(dB

)

0 200 400 600 800

0.2

0.4

0.6

0.8

1.0

–200

(a)

–0.4

–0.2

0.0

Frequency (MHz)

Sign

al (

dBm

)

10 100 1000

–150

–140

–130

–120

–110

–160

(b)

Reduced temperature0.35 Tc0.3 Tc0.6 Tc

Reduced temperature0.57 Tc0.9 Tc

Z2338

0

1

2

3

4

94

95

96

97

98

99

100

Vol

tage

(m

V)

Ele

ctro

n te

mpe

ratu

re (

K)

1.57 ps

14-GHzoscilloscope

0

100

200

300

Vol

tage

(mV

)

Time (2 ns/div)

(a)

12

10

8

64

2

0–2

–4–6

–8

T = 60 K

Time (1 ps/div)

Vol

tage

(m

V)

(b)



spot, the nonequilibrium quasiparticle concentration increasesdue to hot-electron thermalization (multiplication) and de-creases due to electron out-diffusion. The normal spot at theabsorption site occurs when the concentration of nonequilibriumelectrons exceeds the critical value corresponding to the localnormal state. If the film thickness d is small compared to Lth,the concentration of nonequilibrium thermalized quasipar-ticles is given by

∂∂

= ∇ + + ( )t

n D nn d

dtM tq q

q

e∆ ∆

∆2

τ, (8)

where M(t) is the multiplication factor and D is the normal-state electron diffusivity. The maximum value that M(t) reachesduring the avalanche multiplication process is called quantumyield or quantum gain; it is proportional to the energy of theabsorbed quantum. Under assumptions that the M(t) rate ismuch larger than the 1/τe rate and that the photon is absorbedat t = 0 and r = 0, the solution for the time-dependent quasipar-ticle concentration profile takes the form

∆n r tM t

Dd te eq

t r Dte, .( ) = ( ) − −4

1 2 4

πτ (9)

The diameter of the normal spot is determined from thecondition n n r t nq q0 0( ) + ( ) >∆ , . The maximum diameter ofthe normal spot increases with the quantum energy. The model28

predicts an almost-Gaussian response pulse with a magnitudethat, up to a certain extent, does not depend on the photonenergy. On the other hand, the pulse duration is a function of themaximum spot size, providing the basis for spectral sensitivityof the device. Finally, the single-quantum detection regimeshould have a cutoff wavelength that depends on operatingconditions (bias current and temperature) and the detector size.Since such a detector counts individual photons, it should haveultimate background-limited sensitivity through the wholerange of operation conditions.

Gol’tsman et al.29 experimentally demonstrated the super-current-assisted, hot-spot-detection mechanism for single op-tical (790-µm-wavelength) photons. Figure 87.35 shows acollection of “snapshots” recorded by a 1-GHz-bandwidthoscilloscope for different energies per laser pulse, incident onthe NbN quantum HEP. Each snapshot presents an 80-ns-longrecord of the response to six successive 100-fs-wide pulses andwas randomly selected out of a real-time detector output datastream. Trace A in Fig. 87.35 corresponds to an average of 100

Figure 87.35Response of a NbN quantum detector to trains of 100-fs optical pulses witha different number of photons per pulse (see text for details).

photons per pulse hitting the detector. In this case, the HEPresponded to each optical pulse in the laser train. The same100%-efficient response was observed (trace B) when therewere approximately 50 photons per pulse. As the incident laserintensity was further decreased (with other experimental con-ditions unchanged), the quantum nature of the detector re-sponse emerged. Instead of the linear decrease of the signalamplitude with incident light intensity, which is characteristicof a classical integrating detector, the response amplitude ofthe single-photon HEP remained nominally the same. In addi-tion, some of the response pulses were missing because of thelimited quantum efficiency of the device as well as fluctuationsin the number of photons incident on the detector. The quantumvoltage response of the HEP is most apparent in the bottom twopairs of traces: C and D (five photons/pulse) and E and F (onephoton/pulse). Each pair corresponds to two different ran-domly selected records obtained under exactly the same ex-perimental conditions. Note that in each case the detectorresponse is very different. Averaging over a long observationtime, however, showed that both the average number of cap-tured pulses and their magnitude remained constant if the pulseenergy was unchanged. This unambiguously demonstrated thesingle-photon operation of the device.

Trace AJin 1 pJ100 w/pulse

Trace B0.5 Jin50 w/pulse

Trace C0.05 Jin5 w/pulse

Trace E0.01 Jin1 w/pulse

Time (20 ns/div)

Vol

tage

(20

0 m

V/d

iv)

Trace F0.01 Jin1 w/pulse

Trace D0.05 Jin5 w/pulse

Z2512



For a mean number of photons per pulse (m), the probabilityP(n) of absorbing n photons from a given pulse is proportionalto

P ne m

n

m n

( ) ( )−~

!. (10)

When the mean number of photons m << 1 (achieved, forexample, by attenuating the radiation fluence to reduce thetotal number of photons incident on the detector to an averageof much less than one photon per pulse),

P nm

n

n

( ) ~!

. (11)

Consequently, for very weak photon fluxes, the probability ofdetecting one photon, two photons, three photons, etc., is

P m Pm

Pm

1 22

36

2 3

( ) ( ) ( )~ , ~ , ~ , etc. (12)

Figure 87.36 plots the probability of the detector producingan output voltage pulse as a function of the number of photonsper pulse, incident on the device area for two different valuesof the bias current. The left vertical axis indicates the meannumber of detector counts per second. The right vertical axiscorresponds to the probability of detecting an optical pulse.Open squares correspond to the bias current 0.92 Ic, where Icis the critical current at the operation temperature. Saturationoccurs at high incident photon fluxes. For smaller fluxes, aspredicted by Eq. (11), the experimental data show the lineardecrease of detection probability with the average number ofincident photons over four orders of magnitude, clearly dem-onstrating the single-photon detection. At very low photondoses, experimental data points saturate at the level of0.4-s−1 counts (probability 4 × 10−4) since the experiment wasperformed in an optically unshielded environment. This levelis regarded as the laboratory photon background. The solidsquares in Fig. 87.36 correspond to the same device, operatedunder the same conditions as those for the solid-square data,but biased with 0.8 Ic. Experimental data points now follow aquadratic dependence of detection probability [see Eq. (12)],showing the two-photon detection. As expected for a two-photon process, the quantum efficiency is significantly lowerthan for the single-photon detection. At the same time, photonbackground is no longer observed since the probability of two

Figure 87.36Count rates and the corresponding counting probability for a NbN quantumdetector as a function of the radiation intensity. Depending on bias current, thedetector can count single-photon (red squares) or two-photon (blue squares)events (Ref. 29).

uncorrelated, stray photons hitting the device within its re-sponse duration is negligibly small.

A nonequilibrium model of a single quantum x-ray detectorwith the readout via the superconducting tunneling junctionwas developed by Twerenbold.30 Typically, a tunnel-junctiondetector consists of a relatively thick absorber film with anunderlying thinner trapping layer, which forms one junctionelectrode. A photon captured in the absorber generates a high-energy photoelectron that relaxes via hot-electron multiplica-tion into the energy gap of the absorber. Nonequilibriumquasiparticles excited during the cascade diffuse to the adja-cent trapping layer, which has a smaller energy gap. There,quasiparticles scatter inelastically, reaching an energy levelcorresponding to the trapping-layer energy gap. The latterprocess is called “trapping” because it confines the charge tothe region close to the tunnel barrier. The tunnel junction isexternally biased in such a way that trapped quasiparticles cantunnel directly to the electrode characterized by the lower-energy gap. The same potential barrier prevents them fromreturning. They can, however, break Cooper pairs in the low-gap electrode and then form new pairs with unpaired electronsin their own electrode. Thus, the process returns unpairedelectrons to the initial electrode, increasing the number oftunneling events per quasiparticle and providing intrinsiccharge amplification. The time integral of the current transient

1000

100

10

1

0.1

0.01

0.00110–6 10–4 1001

10–6

10–4

10–2

100

Prob

abili

ty

Average number of photons per pulseincident on the device

Cou

nts

per

seco

nd

Z2513

10–5

10–3

10–1

10–2

Ibias = 0.92 Icm1 dependence

Ibias = 0.8 Icm2 dependence



gives, with no free parameters, the charge that has beentransferred through the tunnel junction. This latter value isproportional to the number of quasiparticles created in thecascade and, consequently, to the x-ray quantum energy.

The theoretical energy resolution of the tunnel junctiondetector is given by 2 4 1 1 1 2. h F nν ∆ + +( )[ ] , where hν is thequantum energy, n is the number of tunneling events per onequasiparticle, and F is the Fano factor that describes thestatistical fluctuations of the charge-generation process. TheTwerenbold model incorporates the two-dimensional diffu-sion equation for ∆nq and the general nonlinear form of theRT equations.

A more general approach, including time evolution ofnonequilibrium distribution functions of quasiparticles andphonons, was developed by Nussbaumer et al.31 The authorssolved the Chang-Scalapino equations numerically for thequasiparticle and phonon distribution functions in a spatiallyhomogeneous situation and supplemented the solution by one-dimensional diffusion. The full theory includes the parametersthat are important for the real detector, such as back tunnelingand losses of quasiparticles at the edges of the device, resultingin good agreement between the calculated transient responsesignals and the experimentally measured pulse shapes.

Hot-Electron DetectorsA minor, but physically very important, difference exists

between a superconducting HEP and a conventional supercon-ducting bolometer when they are operated in the transition-edge regime. In the bolometer, thermal equilibrium betweenelectrons and phonons is established instantly, whereas in thehot-electron detector these two systems are not in equilibrium.In this review, we restrict ourselves to publications where thenonequilibrium state between the electron and phonon sub-systems was clearly observed. Basically, there are two ways todecouple electrons from phonons: nonequilibrium phononsshould leave the detector at a time scale that is short comparedto τpe, or the intensity of external radiation should vary fasterthan 1/τpe. Depending on the superconductor and experimentalarrangement, a real hot-electron detector falls somewherebetween these two extremes.

1. Transition-Edge Superconducting DetectorsHistorically, the first HEP’s were developed and studied in

the early 1980s by Gershenzon et al.,32 using ultrathin Nbfilms as the detector body. Niobium is characterized by rela-tively long τpe, typically a few hundred nanoseconds at liquidhelium temperature, so that τes < τep for films thinner than

10 nm.11 Therefore, detectors based on thin Nb films belongto the first limiting case in that their response time is approxi-mately equal to τep. The best performance that the Nb HEP’scan achieve33 is NEP = 3 × 10−13 W/Hz1/2, detectivity D* =4 × 1011 cm s1/2 J−1, and a response time of 4.5 ns. Thus, thesedevices are less sensitive, although much faster, than semicon-ductor bolometers. When the detector area was adjusted prop-erly, Nb HEP’s demonstrated a constant value of sensitivity inthe range from microwaves (150 GHz) to ultraviolet (1015 Hz).This is actually their greatest advantage when compared tosemiconductor counterparts. A Nb-based HEP was imple-mented to study the emission of a cyclotron p-germaniumlaser.34 The combination of large sensitivity and short re-sponse time made it possible to identify the Landau levelsresponsible for lasing.

In the late 1990s, the Gershenzon group developed a HEPbased on NbN superconducting films.35 NbN has much shorterτep and τpe than Nb; thus, even for 3-nm-thick films, NbNHEP’s operate in the mixed regime (i.e., τep and τes jointlydetermine the response time of the detector). Detectors madefrom ultrathin NbN films are much faster than Nb-baseddevices. The intrinsic τep ≈ 10 ps, while the overall responsetime is about 30 ps near Tc.

8 The best-demonstrated NEP ≈10−12 W/Hz1/2 (Ref. 36). In spite of a rather-complicatedelectronic band structure,37 the quantum yield in NbN reachesabove 300 for near-infrared photons,38 which corresponds toone-third of the upper theoretical limit. Detectors fabricatedfrom NbN were used to study the emission of optically pumped

Figure 87.37Pulses from a single-shot, optically pumped, far-infrared gas laser recordedwith a NbN HEP (Ref. 39). The inset shows one of the pulses on an expendedtime scale.

50

25

0

0 50 100

Det

ecto

r si

gnal

(m

V)

70 80

Time (ns)Z2540

nFIR = 33.44 cm–1 22 ns



infrared gas lasers, in particular, pulsed lasers.39 Figure 87.37shows far-infrared laser pulses recorded with a NbN hot-electron detector. The unique combination of response timeand sensitivity made it possible to detect and identify veryweak emission lines.

Miller et al.40 have demonstrated a photon counter based onthe transition-edge, hot-electron, direct detector. The devicewas a 20 × 20-µm2 square of 40-nm-thick tungsten film(Fig. 87.38) having Tc = 80 mK with a transition width of 1 mK.The device was operated at a bath temperature of 40 mK in avoltage-bias regime that maintained the sensor within thetransition region via negative electrothermal feedback.41 Thismode of operation was shown to increase the transition-edgesensor sensitivity and to decrease its time constant toτ α0 1+( )n . Here τ0 is the intrinsic time constant of thesensor, n is the power of the temperature dependence of thethermal conductance between the film and substrate, and α isthe dimensionless sharpness parameter of the superconductingtransition.41 A photon absorbed in the sensor heats the electronsystem above its equilibrium temperature, leading to an in-crease of the sensor’s resistance and, consequently, to thedecrease of the bias current and dissipated joule power. Theintegral of the drop in current (read out by an array of dcSQUID’s) gives the energy absorbed by the sensor with no freeparameters. The detector described in Ref. 40 exhibited a timeconstant of about 60 µs and was able to register 0.3-eV (4-µm-wavelength) single photons with an energy resolution of0.15 eV. To test the detector, the authors observed the planetarynebula NGC 6572, using the 8-in. telescope. The energyresolution was somewhat lower than in the laboratory, al-though it was high enough to detect the strong emission lines.

Figure 87.38(a) Microphotograph of a transition-edge, hot-electron quantum detector and(b) the corresponding equivalent circuit (Ref. 40).

A hot-electron microcalorimeter was developed by Nahumand Martinis.42 In this type of device, photon absorption givesrise to Te in a metal absorber and is measured using the I–Vcharacteristics of a normal-insulator-superconductor tunneljunction, in which part of the absorber forms the normalelectrode. Figure 87.39 shows a schematic of the tested device.The current through the junction was measured with a low-noise dc SQUID. The absorber had an area of 100 × 100 µm2

and was deposited on a silicon nitride membrane. In thisconfiguration, the phonons that escaped from the absorberwere reflected back from the membrane and were furtheravailable for the energy exchange. Thus, the Si3N4 membraneprevented energy loss from the electron subsystem in theabsorber. The microcalorimeter operated at 80 mK with a timeconstant of 15 µs and demonstrated an energy resolution of22 eV for 6-keV photons.

Z2492

SQUID

SiO2

Ag electrode

Auabsorber

NIS junction

Agthermalizer

500 mm

Al electrode

Iheat

Alcontacts

V

Si3 N4membrane

Figure 87.39Detailed schematic of the hot-electron microcalorimeter developed byNahum and Martinis (Ref. 42) (see text for explanation).

In another version, Nahum and Martinis43 proposed amicrobolometer that consisted of a normal metal stripe con-nected to superconducting electrodes (Fig. 87.40). The devicerelied on Andreev reflections of low-energy, thermal quasipar-ticles at the edges of the stripe and on weak electron–phononcoupling at low temperatures. Both effects confined the energydelivered by the photons, providing a large rise of Te. This wassubsequently read out by the superconductor-insulator-normalmetal junction, for which the metal strip formed the normalelectrode. Projected responsivity and NEP of the device withthe Cu absorber operated at 100 mK were about 109 V/W and3 × 10−18 W/Hz1/2, respectively, which is at least a factor of 10better than the performance of any currently available detec-tors. The time constant of the microcalorimeter is determined

(a) (b)

Z2489

SQUIDarray

HEP

Rshunt

Ibias



by the rate of energy transfer from electrons to phonons thatcorresponds to τep at the Fermi level. For the device underconsideration in Ref. 43, the computed response time τ =20 µs. Since, for a bolometer, NEP scales as τ −1/2, the deviceperformance can be further improved by increasing the re-sponse time up to a value only slightly less than that requiredby a specific application.

Voltage (mV)

I (n

A)

Z2490

100 200 300

1

2

0

JunctionAluminumelectrode

Copperabsorber

Leadcontacts

I

Ib

V

0

V

Figure 87.40A hot-electron microbolometer using Andreev reflections of quasiparticlesfrom superconducting contacts and the corresponding I–V characteristics(Ref. 43).

Finally, Karasik et al.44 proposed the use of the dependenceof the electron–phonon scattering time on the electron meanfree path to control the intrinsic response time of a transition-edge detector. Increase of the intrinsic response time results inthe decrease of the minimum detectable power, while at thesame time, the device response time can be decreased to areasonable value by exploiting the negative electrothermalfeedback. According to estimates in Ref. 44, using this ap-proach, a detector could be fabricated with NEP = 10−20

W/Hz1/2 and the millisecond τ at 100-mK bath temperature.

2. Superconducting Kinetic-Inductive DetectorsThe detectors described in the preceding section produce a

response when the device, or at least part of it, is in the resistivestate. Kinetic-inductive integrating detectors represent theirsuperconducting counterpart. The Lkin [see Eq. (7)] of a su-perconducting film makes it possible to monitor the concentra-tion of Cooper pairs. In a constant current-biased super-conducting film, after the destruction of a certain number ofCooper pairs, the remaining pairs accelerate to carry the samebias current. Because of non-zero inertia of pairs, or Lkin,acceleration requires an electric field. This intrinsically gener-

ated electric field is seen from the exterior as a voltage pulsedeveloping across the film. Mathematically, this voltage tran-sient is given by

V IdL

dtkinkin= . (13)

Figure 87.34(b) presented earlier the Vkin transient, recordedfor a YBCO microbridge excited by 100-fs optical pulses. Thenumerical fit was based on Eq. (13) and either Eqs. (1) or (6).

The main advantage of superconducting kinetic-inductivedetectors is their low noise power. To realize this advantage, aSQUID readout should be used. Grossman et al.45 describedthe design of a kinetic-inductive detector/mixer with an esti-mated NEP = 2.5 × 10−17 W/Hz1/2 and a bandwidth of5.5 MHz at 100 mK. Unfortunately, a laboratory prototypeshowed only NEP = 4.4 × 10−11 W/Hz1/2 (Ref. 46). Sergeevand Reizer47 performed thorough calculations for both s-waveand d-wave superconductors, including the appropriate quasi-particle distribution function and scattering times. They foundNEP and D* close to those reported in Ref. 45. Bluzer23

proposed a balanced-bias scheme for a kinetic-inductive pho-todetector with directly coupled SQUID readout, intended toeliminate the losses inherent in inductively coupled readoutsand increase the responsivity of the detector. Performance ofthe detector was simulated for a 0.1-µm-thick YBCO film at9 K, resulting in NEP = 2.5 × 10−15 W/Hz1/2 and 10-µsresponse time. It is believed that the use of a LTS materialshould result in a two- to three-orders-of-magnitude decreasein NEP.

3. Superconducting Quantum DetectorsA number of novel approaches proposed during the last

decade have been aimed at the realization of detectors withultimate quantum sensitivity. Kadin and Johnson26 introducedthe quantum detection regime in ultrathin resistive films. In theproposed mechanism, an absorbed photon induces a resistivehot spot, centered at the point where the photon hits the film.If the photon flux is sufficiently low, hot spots do not overlapuntil they disappear. Using material parameters of NbN, theauthors estimated that a 0.1-µm2 size sample should respond to1-eV photons with 1-mV-amplitude pulses and 10-GHz band-width. For technological reasons, practical detectors wouldrequire significantly larger areas and, consequently, muchsmaller responsivities, forcing the implementation of a sophis-ticated readout scheme such as an array of SQUID’s.27



A photon counter using the quantum detection regime in acurrent-carrying superconducting film28 was recently demon-strated by Gol’tsman et al.29 The counting element consistedof a 1.3-µm-long, 0.2-µm-wide microbridge, formed from a6-nm-thick NbN film deposited on a sapphire substrate. Thedetector was operated at 4.2 K, with a bias current of approxi-mately 90% of Ic. Voltage pulses generated by the bridge inresponse to absorbed photons were further amplified by acooled, low-noise amplifier (see Fig. 87.35). The output pulseswere time limited by electronics and had a duration of approxi-mately 100 ps. The intrinsic dark count rate for the detectorwas measured to be below 0.001 s−1 (probability 10−6), whichcorresponds to zero detected responses over 1000 s when theinput was completely blocked. Table 87.I presents the basicparameters of the device operated at the 790-nm wavelength.Single-photon counting was observed in the photon-wave-length range from 0.4 µm to 2.4 µm.48 We note that the devicerepresents a unique combination of the picosecond responsetime and very high responsivity. These characteristics of NbNHEP’s should lead to their practical implementation in areasranging from free-space satellite communication,49 throughquantum communication and quantum cryptography,50 toultraweak luminescence observations and semiconductor inte-grated circuit testing.51 Another exciting application for thistype of detector can be background-limited direct detectorarrays52 for submillimeter astronomy.

Table 87.I: Experimental performance of a NbNphotodetector at 790 nm.

Response time–intrinsic/measured 10 ps/100 ps

Quantum gain factor 340

A/W responsivity 220 A/W

V/W responsivity 4 × 104 V/W

Device quantum efficiency ~20%

Operating temperature ~4 K

Dark counts per second <0.0001

Device noise temperature ~15 K

The most-advanced superconducting quantum detectorsare tunnel-junction detectors, which are being developed for awide range of applications from materials science and mi-croanalysis to particle physics and astrophysics. Only a fewrecent publications are mentioned here because a full reviewof the activities in this field is beyond the scope of this article.Nb-based tunnel-junction detectors with Al trapping layers

have reached, for photon energies of about 70 eV, an energyresolution of 1.9 eV. This performance is limited by thestatistics of quasiparticle multiplication.53 A typical devicehad an area of 50 × 50 µm2. The smallest-detectable, 0.3-eV(4.1-µm-wavelength) photon energy was achieved withTa-based devices54 since this material has an energy gapsmaller than that of Nb. An energy resolution of 0.19 eV wasdemonstrated for 2.5-eV (0.5-µm-wavelength) photons, usingTa-based devices with an area of 20 × 20 µm2 and 12-µsresponse time.

Hot-Electron MixersHistorically, HEM’s have been divided into two large cat-

egories: lattice- or phonon-cooled13 and diffusion-cooled14

devices. As presented earlier, the physics for these two types ofHEM’s is essentially the same. Both types can be described byEqs. (2) using temperature-dependent parameters and properboundary conditions. The analysis becomes easier, however,when the device is designed to be close to one of two extremes,namely, the lattice- or the diffusion-cooling regime. Typically,lattice-cooled mixers are made from thin films of NbN, whereasdiffusion-cooled devices use Nb or Al.

1. Lattice-Cooled MixersCurrent state-of-the-art NbN technology is capable of rou-

tinely delivering 3.0-nm-thick devices that are 500 × 500 nm2

in size with Tc above 9 K. Near Tc, τpe is close to τes, which isabout 40 ps for 3-nm-thick film [see Fig. 87.30(b)]. The τep at8 K is below 20 ps, which results, with the diffusivity of0.5 cm2s−1, in a thermal healing length of about 30 nm. Sincethe device length is typically much larger, the mixer operatesin the phonon-cooled regime. The mixer’s intrinsic IF band-width is determined by the combination of τep and τes timeconstants. In real devices, however, the measured bandwidthdepends strongly on the bias regime. This makes it difficult tocompare published data and reach meaningful conclusions.For HEM’s on Si substrates, the best reported gain and noisebandwidths are 3.5 GHz55 and 8 GHz,56 respectively. Furtherincreases in the bandwidth for lattice-cooled HEM’s can beachieved by using a substrate material that is better thermallycoupled to the superconducting film. One promising candidateis MgO. Recent measurements have shown57 that MgO pro-vides, for a 3.5-nm-thick bolometer, a 4.8-GHz gain bandwidthand 5.6-GHz noise bandwidth, respectively. Further progressin increasing the bandwidth may be achieved by decreasing thebolometer thickness. Recently a 9-GHz gain bandwidth wasreported58 for a 2.5-nm-thick device on MgO. Unfortunately,this direction is limited because NbN films thinner than 2.5 nmbecome inhomogeneous and lose their superconductivity.59



A waveguide version of the receiver with the lattice-cooledNbN HEM has been installed and operated successfully in thefrequency range of 0.6 to 0.8 THz60 and 1.04 THz61 at the10-m Sub-mm Telescope Facility on Mount Graham in Ari-zona. At this telescope, the measured noise temperature of thereceiver was 560 K at 0.84 THz and 1600 K at 1.035 THzover a 1-GHz IF bandwidth centered at 1.8 GHz. The receiverwas used to detect the CO molecular line emission in theOrion nebula (Fig. 87.41). It is worth noting that this was thefirst ground-based observation at a frequency above 1 THz. Aquasi-optical version of the HEM receiver for the THz range iscurrently under preparation for test flights on a stratosphericairplane observatory.62 The mixer will be incorporated into aplanar logarithmic spiral antenna (Fig. 87.42), which is inte-grated with an extended hyperhemispherical silicon lens.

Practical advantages of the lattice-cooled devices are theirstability and the weak sensitivity of their noise temperature tooperation parameters. Figure 87.43 shows that, indeed, thenoise temperature of a NbN hot-electron mixer does not varynoticeably over a broad range of LO power and bias voltage.63

2. Diffusion-Cooled MixersThe bulk of diffusion-cooled mixers has been realized

based on Nb films. At a 4.2-K bath temperature, the 10-nm-thick Nb film typically has τep of about 1 ns and a diffusivity

of 2 cm2s−1,11 which results in Lth ≈ 0.15 µm. Therefore, Nbdevices having a length of 0.1 µm or less operate in thediffusion-cooled regime. It has been shown experimentally64

that the transition to diffusion cooling of electrons occurs at a

Figure 87.41Terahertz CO line in the Orion IRc2 nebula recorded with a NbN hot-electron mixer at a ground-based telescope in Arizona (Ref. 61). The thicksolid line shows a smoothed spectrum at a resolution of 25 MHz. Thetemperature scale of the spectrum is calibrated by taking into account thereceiver noise temperature, the estimated atmospheric opacity, and theestimated efficiency of the telescope.

Figure 87.42Central part of a planar logarithmic spiral antenna with the NbN hot-electron microbridge.

Figure 87.43Double-sideband (DSB) noise temperature of a laboratory heterodynereceiver with NbN HEM at various bias regimes (Ref. 63).

200

0

1.037 1.0365

–100 0 100

Frequency (THz)

Velocity (km/s)

40

0

TM

B (

K)

TA (

K)

*

Z2539

Z2493 500 nm

100

80

60

40

20

00.0 1.0 2.0 3.0

Bias voltage (mV)

Cur

rent

(mA

)

Z2515

T(DSB) < 660 K

660 K < T(DSB) < 800 K

800 K < T(DSB) < 915 K

915 K < T(DSB) < 980 K

980 K < T(DSB) < 1150 K

1150 K < T(DSB) < 1370 K

1370 K < T(DSB) < 1680 K

1680 K < T(DSB) < 1890 K



device length ≈ 0.2 µm. Expected gain bandwidth for a0.1-µm-long device is about 7 GHz, if one assumes uniformelectron heating through the length of the device. Laboratorytests at sub-THz frequencies confirmed theoretical expecta-tions, and a 9-GHz gain bandwidth was measured for a 0.1-µm-long HEM.65 No noise bandwidth data have been reported sofar for diffusion-cooled mixers. Traditionally, quasi-optical,diffusion-cooled HEM’s use a twin-slot or double-dipole pla-nar antenna and a hemispherical lens to couple the LO andsignal radiations to the mixer. The best reported noise tempera-tures for Nb diffusion-cooled mixers are presently almosttwice as large as those of lattice-cooled devices.

Another apparent difference between the two types ofHEM’s is the optimal bias regime, i.e., the regime resulting inthe lowest noise temperature. For a lattice-cooled HEM, theoptimal bias point is within the linear portion of the nonhystereticI–V characteristics,63 whereas optimal operation of diffusion-cooled devices corresponds to the nonlinear portion of ahysteretic I–V curve.65 The difference stems from boundaryconditions imposed on the normal domain. Movement of thedomain walls caused by signal radiation is not influenced bythe contacts66 if they are located far enough from the domainborders. One can envision such a domain as a freestandingdomain in a stable equilibrium state. In the opposite case, whendomain walls are confined near the contacts, the temperatureprofile at the walls slopes more steeply and the wall movementis restricted by the contacts. This hampers the responsivity ofthe HEM. As a result, the length of a diffusion-cooled mixer issmaller than the thermal diffusion length Lth and correspondsto the length of the smallest freestanding domain. Therefore, ina diffusion-cooled HEM, the conversion loss and, conse-quently, the noise temperature are smaller when the domain is“overcooled” and is slightly shorter than the smallest free-standing domain. The actual domain length, as seen from theresistance in the normal state at the optimal operation point,65

is about 0.6 of the mixer physical length, whereas for phonon-cooled HEM’s,63 the domain length is 0.2 of the device length.Since the total noise power at the HEM output is partly due toNyquist noise, smaller responsivity should result in a some-what larger noise temperature. Another disadvantage of thediffusion-cooled HEM is that its hysteretic regime may causeadditional instability67 when accessed by a practical receiver.

For both mixer types, it is common that optimal operation,aimed at the minimal noise temperature, does not provide thelargest-possible IF bandwidth. Both the bandwidth and thenoise temperature increase with the bias current. Thus, varyingthe bias regime allows a compromise between the desired

bandwidth and the noise temperature acceptable for a particu-lar application.

A diffusion-cooled Al mixer has been recently proposed68

as an alternative to Nb devices. Measurements at 30 GHz69

showed that a diffusion-cooled Al mixer exhibits reasonablygood performance, but these data are not conclusive for thedesired THz operation since the quantum energy of 30-GHzphotons remains smaller than the Al energy gap. Moreover,there are concerns19 that Al HEM’s at THz frequencies wouldrequire a large LO power.

Table 87.II and Fig. 87.44 summarize the current state-of-the-art noise temperatures for both the lattice-cooled anddiffusion-cooled HEM’s. The rapid increase in noise tempera-ture with frequency is inconsistent with the hot-electron model.The model suggests that the noise temperature, when correctedfor optical losses, should not depend on frequency unless itapproaches the quantum-limited value h kBν . A proper ac-count of losses in coupling optics does not eliminate the abovediscrepancy; the noise temperature of the mixer alone in-creases with frequency, following closely the 10 h kBν law inthe frequency range from 0.6 THz to 5.2 THz. It has beenshown recently64 that the nonuniform distribution of the high-frequency current across the device may account for this effect.

Figure 87.44Best double-sideband (DSB) noise temperatures for various types of super-conducting hot-electron mixers as a function of signal frequency. The solidline is the hot-electron model prediction.

0

2000

4000

6000

8000

10000

0 1000 2000 3000 4000 5000 6000

Frequency (GHz)

DSB

noi

se te

mpe

ratu

re (K

)

Z2339

10× (quantumnoise limit)

Quasi-opticalWaveguideQuasi-opticalWaveguide

Diffusioncooling

Phononcooling



Table 87.II: Best double-sideband (DSB) noise temperatures reported in the literature for lattice-cooledand diffusion-cooled mixers.

Lattice-cooled mixers

Quasi-optical layout Waveguide layout

Frequency(GHz)

DSB noisetemperature (K) Reference

Frequency(GHz)


620 500 70 430 410 73

750 600 65 636 483 73

910 850 65 840 490 61

1100 1250 65 1017 750 61

1560 1000 71 1030 800 61

1620 700 58 1260 1100 61

2240 2200 71

2500 1100 58

3100 4000 72

4300 5600 72

5200 8800 72

Diffusion-cooled mixers

Quasi-optical layout Waveguide layout

Frequency(GHz)


Frequency(GHz)


630 470 64 530 650 76

1100 1670 74 700 1100 17

1267 1880 75

2500 1800 64

In Fig. 87.45, simulated frequency dependence of the con-version efficiency is compared with the noise temperaturecorrected for optical losses. Good agreement between theexperimental and theoretical results up to 4 THz suggests thatthe increase in the noise temperature should be less pro-nounced for narrower HEM’s.

ConclusionsSuperconductor hot-electron radiation sensors, operated as

either THz-frequency mixers or optical single-photon detec-tors, promise a revolutionary approach for diagnostics, radioastronomy, and quantum cryptography and communications.The unique performance of these devices in heterodyne as wellas in the direct-detection regime results from a combination ofthe hot-electron phenomenon with the high sensitivity of asuperconductor to nonequilibrium electronic states. To takefull advantage of this combination, devices are routinely fab-ricated from ultrathin superconducting films and feature sub-micron lateral dimensions. They are also operated in thevery-low-noise cryogenic environment.

HEM’s proved their reliability and advantageous featuresduring a two-year test on a ground-based telescope. In thefrequency range from 1 THz to 5.2 THz, HEM’s outperformedSchottky diodes, making them the device-of-choice for THzastronomy and communications.

HEP’s demonstrated excellent performance in the spectralrange from far-infrared wavelengths to x rays when operated ineither integrating or quantum regimes. Their future applica-tions are expected in areas ranging from background-limiteddetector arrays for submillimeter astronomy and x-ray spec-troscopy, through practical, high-speed quantum cryptogra-phy, to digital integrated-circuit diagnostics.

ACKNOWLEDGMENTThe authors thank S. Cherednichenko for helpful discussions and

S. Svechnikov for systemizing published data. This research was madepossible in part by Award No. RE-2227 of the U.S. Civilian Research andDevelopment Foundation for the Independent States of the Former SovietUnion. G. N. Gol’tsman acknowledges support of the INTAS (#97-1453,#99-569). R. Sobolewski acknowledges support of the U.S. Office of NavalResearch grant N00014-00-1-0237.



Figure 87.45Frequency dependence of the noise temperature (circles) and conversionlosses (squares) of a NbN HEM (Ref. 68). The solid line shows the calculatedconversion losses that account for the skin effect in the device. The dashedline represents quantum-limited noise temperature hν/kB. The scale of theright axis was adjusted to match calculated conversion losses and correctednoise temperature.

REFERENCES

1. E. M. Conwell, High Field Transport in Semiconductors, Solid StatePhysics Supplement 9 (Academic Press, New York, 1967).

2. M. L. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Sov. Phys.-JETP 4,173 (1957).

3. V. A. Schklovski, [Sov. Phys.] FTT 17, 3076 (1975).

4. N. Perrin and C. Vanneste, Phys. Rev B 28, 5150 (1983).

5. A. D. Semenov et al., Phys. Rev. B 52, 581 (1995).

6. E. M. Gershenzon et al., Sov. Phys.-JETP 59, 442 (1984).

7. M. Lindgren et al., Appl. Phys. Lett. 65, 3398 (1994).

8. K. S. Il’in, M. Lindgren, M. Currie, A. D. Semenov, G. N. Gol’tsman,R. Sobolewski, S. I. Cherednichenko, and E. M. Gershenzon, Appl.Phys. Lett. 76, 2752 (2000).

9. M. Lindgren, M. Currie, C. Williams, T. Y. Hsiang, P. M. Fauchet,R. Sobolewski, S. H. Moffat, R. A. Hughes, J. S. Preston, and F. A.Hegmann, Appl. Phys. Lett. 74, 853 (1999).

10. R. Sobolewski, in Superconducting and Related Oxides: Physics andNanoengineering III, edited by D. Pavuna and I. Bozovic (SPIE,Bellingham, WA, 1998), Vol. 3481, pp. 480–491.

11. E. M. Gershenzon et al., Sov. Phys.-JETP 70, 505 (1990).

12. E. M. Gershenzon et al., Sov. Phys.-Tech. Phys. 34, 195 (1989).

13. E. M. Gershenzon et al., Supercond., Phys. Chem. Technol. 3,1582 (1990); B. S. Karasik and A. I. Elantiev, Appl. Phys. Lett. 68,853 (1996).

14. D. E. Prober, Appl. Phys. Lett. 62, 2119 (1993).

15. R. S. Nebosis et al., in Proceedings of the Seventh InternationalSymposium on Space Terahertz Technology (University of Virginia,Charlottesville, VA, 1996), pp. 601–613.

16. B. S. Karasik, W. R. McGrath, and M. C. Gaidis, J. Appl. Phys. 81,1581 (1997).

17. W. D. Floet et al., in Proceedings of the Tenth International Symposiumon Space Terahertz Technology (University of Virginia, Charlottesville,VA, 1999), pp. 228–236.

18. H. Merkel et al., IEEE Trans. Appl. Supercond. 9, 4201 (1999).

19. A. D. Semenov and G. N. Gol’tsman, J. Appl. Phys. 87, 502 (2000).

20. H. Ekström et al., IEEE Trans. Microw. Theory Tech. 43, 938 (1995).

21. A. Rothwarf and B. N. Taylor, Phys. Rev. Lett. 19, 27 (1967).

22. E. N. Grossman, D. G. McDonald, and J. E. Sauvageau, IEEE Trans.Magn. 27, 2677 (1991).

23. N. Bluzer, J. Appl. Phys. 78, 7340 (1995).

24. A. D. Semenov et al., IEEE Trans. Appl. Supercond. 7, 3083 (1997).

25. M. Lindgren, M. Currie, C. A. Williams, T. Y. Hsiang, P. M. Fauchet,R. Sobolewski, S. H. Moffat, R. A. Hughes, J. S. Preston, and F. A.Hegmann, IEEE J. Sel. Top. Quantum Electron. 2, 668 (1996).

26. A. M. Kadin and M. W. Johnson, Appl. Phys. Lett. 69, 3938 (1996).

27. D. Gupta and A. M. Kadin, IEEE Trans. Appl. Supercond. 9,4487 (1999).

28. A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, Physica C 351,349 (2001).

29. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov,K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski,Appl. Phys. Lett. 79, 705 (2001).

30. D. Twerenbold, Phys. Rev. B 34, 7748 (1986).

31. Th. Nussbaumer et al., Phys. Rev. B 61, 9719 (2000).

32. E. M. Gershenzon et al., JETP Lett. 34, 268 (1981).

33. E. M. Gershenzon et al., Sov. Tech. Phys. Lett. 15, 118 (1989).

34. A. V. Bespalov et al., Solid State Commun. 80, 503 (1991).

35. B. M. Voronov et al., Supercond., Phys. Chem. Technol. 5, 960 (1992).

36. Yu. P. Gousev et al., J. Appl. Phys. 75, 3695 (1994).

0

2000

4000

6000

8000

10000

0 2 4 60

20

40

60

80

Z2538

hn/kB

Frequency (THz)

Noi

se te

mpe

ratu

re (

K)

Con

vers

ion

loss

es (

arbi

trar

y un

its)



37. S. V. Vonsovskii, IU. A. Iziumov, and E. Z. Kurmaev, Superconductiv-ity of Transition Metals: Their Alloys and Compounds, Springer Seriesin Solid-State Sciences (Springer-Verlag, Berlin, 1982).

38. K. S. Il’in, I. I. Milostnaya, A. A. Verevkin, G. N. Gol’tsman, E. M.Gershenzon, and R. Sobolewski, Appl. Phys. Lett. 73, 3938 (1998).

39. P. T. Lang et al., Appl. Phys. B B53, 207 (1991); P. T. Lang et al., Opt.Lett. 17, 502 (1992).

40. A. J. Miller et al., IEEE Trans. Appl. Supercond. 9, 4205 (1999).

41. K. D. Irwin, Appl. Phys. Lett. 66, 1998 (1995).

42. M. Nahum and J. M. Martinis, Appl. Phys. Lett. 66, 3203 (1995); K. D.Irwin et al., Nucl. Instrum. Methods Phys. Res. A 444, 184 (2000).

43. M. Nahum and J. M. Martinis, Appl. Phys. Lett. 63, 3075 (1993).

44. B. S. Karasik et al., Supercond. Sci. Technol. 12, 745 (1999).

45. E. N. Grossman, D. G. McDonald, and J. E. Sauvageau, IEEE Trans.Magn. 27, 2677 (1991).

46. J. E. Sauvageau, D. G. McDonald, and E. N. Grossman, IEEE Trans.Magn. 27, 2757 (1991).

47. A. V. Sergeev and M. Yu. Reizer, Int. J. Mod. Phys. B 10, 635 (1996).

48. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Dzardanov,K. Smirnov, A. Semenov, B. Voronov, C. Williams, and R. Sobolewski,IEEE Trans. Appl. Supercond. 11, 574 (2001).

49. G. G. Ortiz, J. V. Sandusky, and A. Biswas, in Free-Space LaserCommunication Technologies XII, edited by G. S. Mercherle (SPIE,Bellingham, WA, 2000), Vol. 3932, pp. 127–138.

50. G. Gilbert and M. Hamrick, “Practical Quantum Cryptography: AComprehensive Analysis (Part One),” to appear in Phys. Rep.; see alsohttp://xxx.lanl.gov/abs/quant-ph/0009027.

51. J. C. Tsang and J. A. Kash, Appl. Phys. Lett. 70, 889 (1997).

52. G. Gol’tsman et al., “Background Limited Quantum SuperconductingDetector for Submillimeter Wavelengths,” to be published in theProceedings of the Twelfth International Symposium on Space TerahertzTechnology.

53. S. Friedrich et al., IEEE Trans. Appl. Supercond. 9, 3330 (1999).

54. P. Verhoeve et al., IEEE Trans. Appl. Supercond. 7, 3359 (1997).

55. S. Cherednichenko et al., in Proceedings of the Eighth InternationalSymposium on Space Terahertz Technology (Harvard University, Cam-bridge, MA, 1997), pp. 245–252.

56. H. Ekström et al., Appl. Phys. Lett. 70, 3296 (1997).

57. S. Cherednichenko et al., in Proceedings of the Eleventh InternationalSymposium on Space Terahertz Technology (Ann Arbor, MI, 2000),pp. 219–227.

58. M. Kroug et al., IEEE Trans. Appl. Supercond. 11, 962–965 (2001).

59. B. M. Voronov, Moscow State Pedagogical University, private com-munication (2000).

60. J. H. Kawamura et al., IEEE Trans. Appl. Supercond. 9, 3753 (1999);J. H. Kawamura et al., Publ. Astron. Soc. Pac. 111, 1088 (1999).

61. C.-Y. E. Tong et al., in Proceedings of the Eleventh InternationalSymposium on Space Terahertz Technology, (Ann Arbor, MI, 2000),pp. 49–59.

62. H.-W. Huebers et al., in Airborne Telescope Systems, edited by R. K.Melugin and H.-P. Roeser (SPIE, Bellingham, WA, 2000), Vol. 4014,pp. 195–202.

63. P. Yagoubov et al., Appl. Phys. Lett. 73, 2814 (1998).

64. P. J. Burke et al., Appl. Phys. Lett. 68, 3344 (1996).

65. R. A. Wyss et al., in Proceedings of the Tenth International Symposiumon Space Terahertz Technology (University of Virginia, Charlottesville,VA, 1999), pp. 215–228.

66. A. D. Semenov and H.-W. Hubers, IEEE Trans. Appl. Supercond. 11,196 (2001).

67. A. D. Semenov et al., J. Appl. Phys. 88, 6758 (2000).

68. B. S. Karasik and W. R. McGrath, in Proceedings of the NinthInternational Symposium on Space Terahertz Technology (1998),p. 73.

69. I. Siddiqi et al., in Proceedings of the Eleventh International Sympo-sium on Space Terahertz Technology, (Ann Arbor, MI, 2000),pp. 82–94.

70. P. Yagoubov et al., Supercond. Sci. Technol. 12, 989 (1999).

71. E. Gerecht et al., in Proceedings of the Tenth International Symposiumon Space Terahertz Technology (University of Virginia, Charlottesville,VA, 1999), pp. 200–207.

72. J. Schubert et al., Supercond. Sci. Technol. 12, 748 (1999).

73. J. Kawamura et al., in Proceedings of the Eighth International Sympo-sium on Space Terahertz Technology (Harvard University, Cambridge,MA, 1997), pp. 23–26.

74. A. Skalare et al., in Proceedings of the Ninth International Symposiumon Space Terahertz Technology (1998), pp. 115–120.

75. A. Skalare et al., IEEE Trans. Appl. Supercond. 7, 3568 (1997).

76. A. Skalare et al., Appl. Phys. Lett. 68, 1558 (1996).

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